LAB4

# Lab 4: Distributions in R

# 1. Uniform Distribution

# a. Generate a Sample
# Generate 500 random numbers from U(0, 10)
uniform_sample <- runif(500, min = 0, max = 10)

# Summary and visualization
summary(uniform_sample)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.04021 2.37882 4.72349 4.85390 7.26791 9.98140

hist(uniform_sample, main = "Uniform Distribution (0, 10)", xlab = "Values", col = "lightblue")

# b. Calculate Probability
# CDF for uniform distribution: P(X <= 4)
p_uniform <- punif(4, min = 0, max = 10)
print(p_uniform)

## [1] 0.4

# 2. Exponential Distribution

# a. Generate a Sample
# Generate 500 random numbers from Exp(0.1)
exponential_sample <- rexp(500, rate = 0.1)

# Summary and visualization
summary(exponential_sample)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.0014  2.8979  7.4235 10.2519 13.6548 73.0231

hist(exponential_sample, main = "Exponential Distribution (rate = 0.1)", xlab = "Values", col = "lightgreen")

# b. Calculate Probability
# Survival function: P(X > 7)
p_exponential <- 1 - pexp(7, rate = 0.1)
print(p_exponential)

## [1] 0.4965853

# 3. Poisson Distribution

# a. Generate a Sample
# Generate 500 random numbers from Poisson(3)
poisson_sample <- rpois(500, lambda = 3)

# Summary and visualization
summary(poisson_sample)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   2.000   3.000   3.044   4.000  11.000

hist(poisson_sample, main = "Poisson Distribution (lambda = 3)", xlab = "Values", col = "lightcoral")

# b. Calculate Probability
# Probability of exactly 4 events: P(X = 4)
p_poisson <- dpois(4, lambda = 3)
print(p_poisson)

## [1] 0.1680314

# 4. Binomial Distribution

# a. Generate a Sample
# Generate 500 random numbers from Binomial(10, 0.5)
binomial_sample <- rbinom(500, size = 10, prob = 0.5)

# Summary and visualization
summary(binomial_sample)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   4.000   5.000   4.958   6.000   9.000

hist(binomial_sample, main = "Binomial Distribution (n = 10, p = 0.5)", xlab = "Values", col = "lightyellow")

# b. Calculate Probability
# Probability of exactly 3 successes: P(X = 3)
p_binomial <- dbinom(3, size = 10, prob = 0.5)
print(p_binomial)

## [1] 0.1171875

# 5. Bonus: Visualization of Probability Distributions

# a. Overlay Histograms
# Overlay the histograms of the uniform, exponential, Poisson, and binomial distributions
hist(uniform_sample, main = "Comparison of Distributions", xlab = "Values", col = rgb(0.2, 0.5, 0.7, 0.5), xlim = c(0, 30), breaks = 15)
hist(exponential_sample, col = rgb(0.3, 0.7, 0.3, 0.5), add = TRUE, breaks = 15)
hist(poisson_sample, col = rgb(0.8, 0.3, 0.3, 0.5), add = TRUE, breaks = 15)
hist(binomial_sample, col = rgb(0.7, 0.7, 0.1, 0.5), add = TRUE, breaks = 15)
legend("topright", legend = c("Uniform", "Exponential", "Poisson", "Binomial"), fill = c(rgb(0.2, 0.5, 0.7, 0.5), rgb(0.3, 0.7, 0.3, 0.5), rgb(0.8, 0.3, 0.3, 0.5), rgb(0.7, 0.7, 0.1, 0.5)))

# Plot density for each distribution

# Set up the plotting area
plot(density(uniform_sample), main = "Comparison of Distributions", xlab = "Values", col = "blue", lwd = 2, xlim = c(0, 30), ylim = c(0, 0.4))
lines(density(exponential_sample), col = "green", lwd = 2)
lines(density(poisson_sample), col = "red", lwd = 2)
lines(density(binomial_sample), col = "yellow", lwd = 2)

# Add a legend to identify the distributions
legend("topright", legend = c("Uniform", "Exponential", "Poisson", "Binomial"), 
       col = c("blue", "green", "red", "yellow"), lwd = 2)

# Load ggplot2 library
install.packages("ggplot2")

## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.4'
## (as 'lib' is unspecified)

library(ggplot2)

# Create data frames for each distribution
uniform_df <- data.frame(values = uniform_sample, distribution = "Uniform")
exponential_df <- data.frame(values = exponential_sample, distribution = "Exponential")
poisson_df <- data.frame(values = poisson_sample, distribution = "Poisson")
binomial_df <- data.frame(values = binomial_sample, distribution = "Binomial")

# Combine the data frames into one
combined_df <- rbind(uniform_df, exponential_df, poisson_df, binomial_df)

# Create the plot with ggplot2
ggplot(combined_df, aes(x = values, fill = distribution)) +
  geom_histogram(alpha = 0.5, position = "identity", bins = 15) + 
  scale_fill_manual(values = c("lightblue", "lightgreen", "lightcoral", "lightyellow")) +
  labs(title = "Comparison of Distributions", x = "Values", y = "Frequency") +
  theme_minimal() +
  theme(legend.title = element_blank())